I am under the impression that there is a set of axioms on which elementary number theory unfolds. If this is true, what are the axioms? Are they the five Peano axioms (at least, thought there were five) for arithmetic.
(This link says there are nine: https://en.wikipedia.org/wiki/Peano_axioms#:~:text=The%20nine%20Peano%20axioms%20contain,the%20set%20of%20natural%20numbers.)
Was elementary number theory built on a set of axioms prior to Peano? Any light you may shed on elementary number theory and its foundation is appreciated.
No, elementary number theory goes back around two thousand years, at least. Books VII-IX of Euclid's Elements concern elementary number theory. And Euclid was a systematizer, collecting results that had been obtained hundreds of years earlier. Some of this material is believed to go back to the Pythagoreans.
Euclid's "arithmetical" books begin with a bunch of definitions, but no postulates are given. The reasoning in his theorems could be translated into modern axiomatic form, but he did not present it that way.
Peano's postulates date from the beginning of the 20th century. Before then, mathematicians used arguments that can be recognized as equivalent. For example, Fermat used the so-called Method of Infinite Descent to prove many results. To prove that all numbers have some property, you begin by assuming that some number is a counter-example, and then prove that there must exist a smaller counterexample. Contradiction, QED. This method is equivalent to using an induction axiom.
Nowadays, the term "Peano's postulates" is used in two different senses, distinguished as the first-order theory and the second-order theory. The first-order theory does not include the general notion of a set of numbers; the induction axiom becomes a so-called axiom schema, with one postulate for each expression $\phi(n)$ (or predicate, as it's called) in a formal language. In the second-order theory, we do have arbitrary sets of numbers. It can also be formalized using the machinery of mathematical logic.
There's a lot more to be said, but I'll quit here.